Linear Algebra

Section IV. Jordan Form 389Jordan form is a canonical form for similarity classes of square matrices,provided that we make it unique by arranging the Jordan blocks from leasteigenvalue to greatest and then arranging the subdiagonal 1 blocks inside eachJordan block from longest to shortest.2.13 Example This matrix has the characteristic polynomial (x − 2) 2 (x − 6).⎛T = ⎝ 2 0 1⎞0 6 2⎠0 0 2We will handle the eigenvalues 2 and 6 separately.Computation of the powers, and the null spaces and nullities, of T − 2I isroutine. (Recall from Example 2.3 the convention of taking T to represent atransformation, here t: C 3 → C 3 , with respect to the standard basis.)power p (T − 2I) p N ((t − 2) p ) nullity⎛ ⎞ ⎛ ⎞0 0 1x⎜ ⎟ ⎜ ⎟1 ⎝0 4 2⎠ { ⎝ 0⎠ ∣ x ∈ C} 1230 0 00⎛ ⎞ ⎛ ⎞0 0 0 x⎜ ⎟ ⎜ ⎟⎝0 16 8⎠ { ⎝−z/2⎠ ∣ x, z ∈ C} 20 0 0 z⎛ ⎞0 0 0⎜ ⎟⎝0 64 32⎠ –same– —0 0 0So the generalized null space N ∞ (t − 2) has dimension two. We’ve noted thatthe restriction of t − 2 is nilpotent on this subspace. From the way that thenullities grow we know that the action of t − 2 on a string basis β ⃗ 1 ↦→ β ⃗ 2 ↦→ ⃗0.Thus the restriction can be represented in the canonical formN 2 =( )0 01 0⎛= Rep B,B (t − 2) B 2 = 〈 ⎝ 1 ⎞ ⎛1 ⎠ , ⎝ −2⎞0 ⎠〉−2 0where many choices of basis are possible. Consequently, the action of the restrictionof t to N ∞ (t − 2) is represented by this matrix.( )2 0J 2 = N 2 + 2I = Rep B2,B 2(t) =1 2The second eigenvalue’s computations are easier. Because the power of x−6in the characteristic polynomial is one, the restriction of t−6 to N ∞ (t−6) mustbe nilpotent of index one. Its action on a string basis must be ⃗ β 3 ↦→ ⃗0 and sinceit is the zero map, its canonical form N 6 is the 1×1 zero matrix. Consequently,